Some important problems and their complexity status

1. Some important problems and their complexity status Mohammad KAYKOBAD Visiting Professor Computer Engineering Department Kyung Hee University CSE Department, North South University

2. Resolved problems • There were 12 important problems whose complexity status was unresolved. Of them 8 cases have been resolved • Most important of them are as follows: • Linear Programming solved in 1979 by LG Khachiyan a Russian mathematician by discovering Ellipsoid Algorithm later on Karmarkar also devised a polynomial time algorithm. This discovery was in full compliance of prediction of theory of computational complexity

3. Solved Problems(contd) • Subgraph Homeomorphism problem solved by Robertson and Seymour(1986) • Spanning Tree Parity Problem solved by Lovasz in 1980 to be polynomial • Total Unimodularity solved by Seymour in 1980 to be polynomial • Graph Genus problem has been shown to be NP-Complete by Thomassen

4. Solved Problems(contd) • Chordal Graph Completion proved to be NP-Complete by Yannakakis in 1981 • Chromatic Index proved to be NP-Complete by Holyer in 1981. • Partial order Dimension proved to be NP-Complete by Yannakakis in 1982. • Crossing number proved to be NP-Complete by Garey and Johnson in 1981.

5. Solved Problems(contd) • Graph Thickness proved to be NP-Complete by Mansfield 1983. • Linear Complementarity proved to be NP-Complete by Chung 1979. • Primality Testing has been shown to be polynomial by M Aggrawal in 2002.

6. Important Unresolved Problems • Graph Isomorphism- The Complexity of Some Isomorphism Problems • Many of the mathematical problems are used in a wide variety of applications, e.g., storing images in JPEG format, storing songs in MP3 format, error-free and secure transmission of data etc.In this writeup, three problems from mathematics are discussed: Graph Isomorphism, F-Algebra Isomorphism, and Cubic Form Equivalence.

7. The Graph Isomorphism problem is as follows. We are given two undirected simple graphs over n vertices, say G and H. The vertices in both the graphs are numbered from 1 to n. The problem is to decide if the two graphs are isomorphic. In other words, to check if renumbering the vertices of G make it equal to H. This problem has had a long history.

8. Unresolved problems (contd) • It is useful in several places, for example in classifying the structure of large molecules (the atoms are "vertices" and bonds between them are "edges"). It is known that this problem is unlikely to be NP-hard and so the problem might be easy to solve. At the same time, no efficient algorithm is known for solving the problem.

9. Unresolved problems (contd) • The F-Algebra Isomorphism problem is as follows. Given two algebras over field F (algebras are commutative rings with identity), test if they are isomorphic. This is one of the basic problems in mathematics. When the two algebras are fields, it is easy to test the isomorphism. However, in general, the problem is not easy to solve.

10. Unresolved problems (contd) • When F is an algebraically closed field (for example, the field of complex numbers), then -- using the famous Hilbert's Nullstellensatz -- one can show that the problem can be solved within polynomial space. When F is the field of reals, then -- using another famous result by Tartski on decidability of first-order theory of reals -- one can show that the problem can be solved in doubly exponential time.

11. Unresolved problems (contd) • When F is the field of rational numbers, the status of the problem is open -- it is not even known to be decidable! When F is a finite field, its complexity is very similar to that of Graph • Isomorphism.The Cubic Form Equivalence problem is as follows. Given two homogeneous, degree three polynomials over field F, test if the first one becomes equal to the second one under a linear transformation. This problem is also very well studied in mathematics. In case of degree two polynomials (instead of degree three), the problem has a very elegant and efficient solution for any F. However, degree three case does not appear so easy.

12. Unresolved problems (contd) • The complexity of the problem also depends on the field F and behaves in the same way as the complexity of the F-Algebra Isomorphism problem. Interestingly, although these three problems do not seem related to each other, it can be shown that, in fact, they are! The Graph Isomorphism problem can be transformed to F-Algebra Isomorphism problem which, in turn, can be transformed to Cubic Form Equivalence problem. (Manindra Agrawal)

13. Unresolved problems (contd) • Still unresolved are problems like graceful labelling of trees- Given an arbitrary tree on n vertices is it possible to label them with distinct integers 0,1,…,n-1 so that edges get labels 1,2,…, n-1, where edge label is absolute difference of vertex labels.

14. Unresolved problems (contd) • Graph Reconstruction Problem asks to reconstruct the graph given n subgraphs induced by the vertices of the original graph removing everytime a different vertex one at a time. The problem has been shown to be polynomially solvable if the graph is a tree.

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